Recall that the condition on the nonisolated singularity at the origin is necessary for the existence of local analytic or formal first integrals.
A pluripolar set is a subset of \(\mathbb {C}^m\)  for some \(m\in \mathbb {N}\) , and it is of Lebesgue measure zero, see e.g. [1]} or [2]}. A detail definition on pluripolar set will be given in Section , where we prove Theorem REF .
